482 research outputs found
Between Poisson and GUE statistics: Role of the Breit-Wigner width
We consider the spectral statistics of the superposition of a random diagonal
matrix and a GUE matrix. By means of two alternative superanalytic approaches,
the coset method and the graded eigenvalue method, we derive the two-level
correlation function and the number variance . The graded
eigenvalue approach leads to an expression for which is valid for all
values of the parameter governing the strength of the GUE admixture
on the unfolded scale. A new twofold integration representation is found which
can be easily evaluated numerically. For the Breit-Wigner width
measured in units of the mean level spacing is much larger than
unity. In this limit, closed analytical expression for and
can be derived by (i) evaluating the double integral
perturbatively or (ii) an ab initio perturbative calculation employing the
coset method. The instructive comparison between both approaches reveals that
random fluctuations of manifest themselves in modifications of the
spectral statistics. The energy scale which determines the deviation of the
statistical properties from GUE behavior is given by . This is
rigorously shown and discussed in great detail. The Breit-Wigner
width itself governs the approach to the Poisson limit for . Our
analytical findings are confirmed by numerical simulations of an ensemble of
matrices, which demonstrate the universal validity of our
results after proper unfolding.Comment: 25 pages, revtex, 5 figures, Postscript file also available at
http://germania.ups-tlse.fr/frah
Norm-dependent Random Matrix Ensembles in External Field and Supersymmetry
The class of norm-dependent Random Matrix Ensembles is studied in the
presence of an external field. The probability density in those ensembles
depends on the trace of the squared random matrices, but is otherwise
arbitrary. An exact mapping to superspace is performed. A transformation
formula is derived which gives the probability density in superspace as a
single integral over the probability density in ordinary space. This is done
for orthogonal, unitary and symplectic symmetry. In the case of unitary
symmetry, some explicit results for the correlation functions are derived.Comment: 19 page
Arbitrary Rotation Invariant Random Matrix Ensembles and Supersymmetry
We generalize the supersymmetry method in Random Matrix Theory to arbitrary
rotation invariant ensembles. Our exact approach further extends a previous
contribution in which we constructed a supersymmetric representation for the
class of norm-dependent Random Matrix Ensembles. Here, we derive a
supersymmetric formulation under very general circumstances. A projector is
identified that provides the mapping of the probability density from ordinary
to superspace. Furthermore, it is demonstrated that setting up the theory in
Fourier superspace has considerable advantages. General and exact expressions
for the correlation functions are given. We also show how the use of hyperbolic
symmetry can be circumvented in the present context in which the non-linear
sigma model is not used. We construct exact supersymmetric integral
representations of the correlation functions for arbitrary positions of the
imaginary increments in the Green functions.Comment: 36 page
The k-Point Random Matrix Kernels Obtained from One-Point Supermatrix Models
The k-point correlation functions of the Gaussian Random Matrix Ensembles are
certain determinants of functions which depend on only two arguments. They are
referred to as kernels, since they are the building blocks of all correlations.
We show that the kernels are obtained, for arbitrary level number, directly
from supermatrix models for one-point functions. More precisely, the generating
functions of the one-point functions are equivalent to the kernels. This is
surprising, because it implies that already the one-point generating function
holds essential information about the k-point correlations. This also
establishes a link to the averaged ratios of spectral determinants, i.e. of
characteristic polynomials
Regularities and Irregularities in Order Flow Data
We identify and analyze statistical regularities and irregularities in the
recent order flow of different NASDAQ stocks, focusing on the positions where
orders are placed in the orderbook. This includes limit orders being placed
outside of the spread, inside the spread and (effective) market orders. We find
that limit order placement inside the spread is strongly determined by the
dynamics of the spread size. Most orders, however, arrive outside of the
spread. While for some stocks order placement on or next to the quotes is
dominating, deeper price levels are more important for other stocks. As market
orders are usually adjusted to the quote volume, the impact of market orders
depends on the orderbook structure, which we find to be quite diverse among the
analyzed stocks as a result of the way limit order placement takes place.Comment: 10 pages, 9 figure
Counting function for a sphere of anisotropic quartz
We calculate the leading Weyl term of the counting function for a
mono-crystalline quartz sphere. In contrast to other studies of counting
functions, the anisotropy of quartz is a crucial element in our investigation.
Hence, we do not obtain a simple analytical form, but we carry out a numerical
evaluation. To this end we employ the Radon transform representation of the
Green's function. We compare our result to a previously measured unique data
set of several tens of thousands of resonances.Comment: 16 pages, 11 figure
Transition from Poisson to gaussian unitary statistics: The two-point correlation function
We consider the Rosenzweig-Porter model of random matrix which interpolates
between Poisson and gaussian unitary statistics and compute exactly the
two-point correlation function. Asymptotic formulas for this function are given
near the Poisson and gaussian limit.Comment: 19 pages, no figure
Invariant Manifolds and Collective Coordinates
We introduce suitable coordinate systems for interacting many-body systems
with invariant manifolds. These are Cartesian in coordinate and momentum space
and chosen such that several components are identically zero for motion on the
invariant manifold. In this sense these coordinates are collective. We make a
connection to Zickendraht's collective coordinates and present certain
configurations of few-body systems where rotations and vibrations decouple from
single-particle motion. These configurations do not depend on details of the
interaction.Comment: 15 pages, 2 EPS-figures, uses psfig.st
Stochastic field theory for a Dirac particle propagating in gauge field disorder
Recent theoretical and numerical developments show analogies between quantum
chromodynamics (QCD) and disordered systems in condensed matter physics. We
study the spectral fluctuations of a Dirac particle propagating in a finite
four dimensional box in the presence of gauge fields. We construct a model
which combines Efetov's approach to disordered systems with the principles of
chiral symmetry and QCD. To this end, the gauge fields are replaced with a
stochastic white noise potential, the gauge field disorder. Effective
supersymmetric non-linear sigma-models are obtained. Spontaneous breaking of
supersymmetry is found. We rigorously derive the equivalent of the Thouless
energy in QCD. Connections to other low-energy effective theories, in
particular the Nambu-Jona-Lasinio model and chiral perturbation theory, are
found.Comment: 4 pages, 1 figur
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